I try to make my question clear:

When reading a paper or listening a seminar talk, people showed me some set, and claim it to be a scheme; or some map, and claim it to be a morphism. I query why this is the case, and the answer sometimes goes as following : because every construction is algebraic, the result should be algebraic.

I don't know if this can be deemed as a proof or something easy to be checked (I am certainly a novice in AG) or a principle like Lefschetz principle. I do find a list of constructions called algebraic construction from Wikipedia. However I do not have any idea of how it is defined, and how to use them in the proof.

If you have heard similar explanation as I did, I hope you could understand my puzzle; and if you have use the similar explanation, I hope you could elaborate a little bit what does this exactly mean.

arbitrary(or finitely generated) $\mathbf{C}$-algebras -- the latter is what is actually needed in rigorous arguments but the notation can get clunky in that generality, so experts omit it; Mumford explains it extremely well in his book on abelian varieties. – user36938 Jul 18 '13 at 14:35